Isotropic and anisotropic hyperfine coupling paramters

Python API and CLI

Use the hyperfine_parameters() function or the CLI tool:

$ python3 -m gpaw.hyperfine <gpw-file>
gpaw.hyperfine.hyperfine_parameters(calc: gpaw.calculator.GPAW, exclude_core=False) → numpy.ndarray[source]

Calculate isotropic and anisotropic hyperfine coupling paramters.

One tensor (\(A_{ij}\)) per atom is returned in eV units. In Hartree atomic units, we have the isotropic part \(a = \text{Tr}(\mathbf{A}) / 3\):

\[a = \frac{2 \alpha^2 g_e m_e}{3 m_p} \int \delta_T(\mathbf{r}) \rho_s(\mathbf{r}) d\mathbf{r},\]

and the anisotropic part \(\mathbf{A} - a\):

\[\frac{\alpha^2 g_e m_e}{4 \pi m_p} \int \frac{3 r_i r_j - \delta_{ij} r^2}{r^5} \rho_s(\mathbf{r}) d\mathbf{r}.\]

Remember to multiply each tensor by the g-factors of the nuclei and divide by the total electron spin.

Use exclude_core=True to exclude contribution from “frozen” core.

For details, see Peter E. Blöchl and Oleg V. Yazyev et al..

G-factors

Here is a list of g-factors (from Wikipedia):

Nucleus

g-factor

\(^{1}\)H

5.586

\(^{3}\)He

-4.255

\(^{7}\)Li

2.171

\(^{13}\)C

1.405

\(^{14}\)N

0.404

\(^{17}\)O

-0.757

\(^{19}\)F

5.254

\(^{23}\)Na

1.477

\(^{27}\)Al

1.457

\(^{29}\)Si

-1.111

\(^{31}\)P

2.261

\(^{57}\)Fe

0.181

\(^{63}\)Cu

1.485

\(^{67}\)Zn

0.350

\(^{129}\)Xe

-1.545

Hydrogen 21 cm line

Here is how to calculate the famous hydrogen spectral line of 21 cm:

import numpy as np
from ase import Atoms
import ase.units as units
from gpaw import GPAW, PW
from gpaw.hyperfine import hyperfine_parameters

h = Atoms('H', magmoms=[1])
h.center(vacuum=3)
h.calc = GPAW(mode=PW(400), txt=None)
e = h.get_potential_energy()
A = hyperfine_parameters(h.calc)[0] * 5.586
a = np.trace(A) / 3
frequency = a * units._e / units._hplanck  # Hz
wavelength = units._c / frequency  # meters
print(f'{wavelength * 100:.1f} cm')

The output will be 23.2 cm. It’s slightly off because the LDA spin-density at the position of the hydrogen nucleus is a bit too low (should be \(1/\pi\) in atomic units).